Roots of f(x) are found numerically by Durand–Kerner iteration: zₖ ← zₖ − f(zₖ)/∏(zₖ−zⱼ), run until convergence, then plotted on the complex plane (Re → x, Im → z).
Gal(f/ℚ) ⊆ Sₙ acts on the n roots.
Solvable in radicals ⟺ Gal(f) is a solvable group
(has an abelian composition series).
- Cubic — discriminant Δ = −4p³−27q² decides it: Δ>0 gives 3 real roots and Gal ≅ C₃ (cyclic, order 3); Δ<0 gives 1 real + 1 conjugate pair and Gal ≅ S₃ (order 6). Both are solvable.
- x⁴−2 — Gal ≅ D₄ (order 8): a 4-cycle "rotate by i" and a reflection "conjugate". Solvable.
- Φ₅(x) — cyclotomic, roots are primitive 5th roots of unity; Gal ≅ (ℤ/5ℤ)* ≅ C₄, generated by ζ↦ζ². Solvable.
- x⁵−2 — Gal ≅ F₂₀ (order 20, metacyclic C₅⋊C₄): "rotate ×ζ" and "ζ↦ζ²". Solvable.
- x⁵−4x−2 — irreducible, 3 real roots + 1 conjugate pair, Gal ≅ S₅ (order 120). S₅ is not solvable (its only proper normal subgroup, A₅, is a non-abelian simple group) — this is the Abel–Ruffini obstruction: no radical formula exists for this quintic.